The strong evidence that the spectral problem of planar N=4 SYM is integrable has led to the discovery of new mathematical structures and calculational techniques far more powerful than standard QFT tools. The recently discovered Quantum Spectral Curve formulates the problem in terms of a system of Q-functions that are interrelated by finite difference equations and have very specific analytical properties. \par At the leading order in perturbation theory, the Q-system is equivalent to that of a psu(2,2$\mid$4) XXX spin chain. Any supersymmetry multiplet of single trace operators in N=4 SYM corresponds to a solution of this Q-system. The presented work aims at systematising the solution for any choice of quantum numbers. \par This procedure has two main challenges: (1) finding the leading order solutions, and (2) iteratively generating perturbative corrections. The canonical way to handle the first is to solve a set of seven coupled Bethe equations. The solution of these algebraic equations is notoriously slow and often leads to many unwanted solutions. We present a new algorithm to efficiently find the leading Q-system. Afterwards, we discuss how this sets the stage for perturbative iterations to any loop order. \\ \\ Based on yet unpublished work with Dmytro Volin.